Question: What is the inverse of the function $h(x)=\dfrac{3}{4}x+12$ ? $h^{-1}(x)=$
Solution: Let's start by replacing $h(x)$ with $y$. $y=\dfrac{3}{4}x+12$ If a function contains the point $(a,b)$, the inverse of that function contains the point $(b,a)$. So if we swap the position of $x$ and $y$ in the equation, we get the inverse relationship. In this case, the function is $y=\dfrac{3}{4}x+12$, so the inverse relationship is $x=\dfrac{3}{4}y+12$. Solving this equation for $y$ will give us an expression for $h^{-1}(x)$. $\begin{aligned} x&=\dfrac{3}{4}y+12\\\\ x-12&=\dfrac{3}{4}y\\\\ \dfrac{4}{3}(x-12)&=y\\\\\\ \end{aligned}$ The inverse of the function is $h^{-1}(x)=\dfrac{4}{3}(x-12)$. [I saw someone solve this problem by originally solving for x. Were they wrong?]